Martingale measure associated with the critical stochastic heat flow
Abstract
In [14], they proved the convergence of the finite dimensional time distribution of the rescaled random fields derived from the discrete stochastic heat equation of -directed polymers in random environment in the critical window. The scaling limit is called critical stochastic heat flow (SHF).
In this paper, we will show that the critical SHF is a continuous semimartingale. Moreover, we will consider the martingale problem associated with the critical SHF in a similar fashion to the super Brownian motion which is one of the well-known measure valued process. Also, we define the martingale measure associated with the critical SHF in the sense of [45, Chapter 2].
The quadratic variation of the martingale measure gives information of the regularity of the critical SHF.
MSC 2020 Subject Classification: Primary 60H17. Secondary 65C35, 60G44.
Key words: Critical stochastic heat flow, Directed polymers in random environment, Martingale problem, Martingale measure.
1 Introduction and main results
Kardar-Parisi-Zhang considered the stochastic partial differential equations (KPZ equation) which describes the evolution of the random interface:
| (KPZ) |
where is space-time white noise on [33]. It is ill-posed due to the term which should be the square of distribution. In [33], they also considered the formal transformation and obtained the multiplicative stochastic heat equation
| (SHE) |
In [3], Bertini and Giacomin studied this transformation mathematically in dimension in the analysis of weakly asymmetric simple exclusion process and SOS process. They considered the mollified stochastic heat equation
| () |
under suitable initial condition , where is a space-mollified noise for the probability density with and . Then, satisfies a mollified KPZ equation
| (KPZε) |
where .
They proved that converges a.s. and in to the solution of (SHE) uniformly on the compact set in . Also, Mueller proved that if is nonnegative, not identically zero, and continuous, then is strictly positive on for all [40]. Thus, we find that also converges to some process a.s., which is so called Cole-Hopf solution of (KPZ).
For KPZ equation, Hairer developed the regularity structures and showed the existence of the distributional solution to (KPZ)[27, 28, 19]. In the singular SPDE literature, the existence of the solution of (KPZ) has been proved by Gubinelli-Imkeller-Perkowski via the Paracontrolled calculus[23], by Gonçalves-Jara via the energy solutions[24], and by Kupiainen via the renormalization group approach[34].
Many attempts have been made to construct solutions of (KPZ) for , where the dimension (, ) is called sub-critical(critical, super-critical) in the literature of singular SPDE[28]. Also, and are called ultraviolet superrenormalizable and infrared renormalizable in the physicists’ language [39], respectively.
One approach is to modify the Bertini-Giacomin’s idea: We consider the stochastic heat equation
| (SHE) |
where and is defined by
In (SHE), the strength of the noise term of () is tuned. Then, it has been proved that there exists a phase transition of the one-point distribution in the following sense. There exists such that if (subcriticale regime), then converges to a non-trivial random variable for each and as , and if (supercritical regime), then vanishes[38, 10]. However, we know that if , then converges to the solution of the non-random heat equation in the weak sense, i.e. for each test function , in probability, where is the Gaussian density with mean and variance [38, 9, 10], which can be regarded as the law of large numbers. Thus, we cannot give the new definition of (SHE) (and hence (KPZ)) for the subcritical regime.
Remark 1.1.
We know that in the subcritical regime, the fluctuations of the centered field and converges to the solution of Edwards-Wilkinson type equation [26, 17, 26, 39, 13, 6, 7, 8, 9, 36, 5] in so-called -regime, which can be regarded as the central limit theorem. For , Junk and Nakajima discussed the fluctuation of the centered field of directed polymers (the discrete counterpart of ()) beyond the -regime[31, 32, 29].
One is interested in the critical case for is the interesting phase. In [2], Bertini and Cancrini retake and consider the critical window around the critical point given by
They proved that if , then the variance of converges to the nontrivial quantity which has the same form as (2.13). Thus, the tightness of the random field follows. Moreover, the finiteness of the higher moments has been verified by Caravenna-Sun-Zygouras[12] and Gu-Quastel-Tsai[25] so that the limit point filed should be random field. Finally, the weak convergence of the random field was proved by Caravenna-Sun-Zygouras for directed polymers setting in finite dimensional time distribution sense in [14] and by Tsai for (SHE) in process level in [44]. Thus, the limit can be regarded as the solution to (SHE) for .
Our main results concern the martingale part of (SHE) for . If we rewrite (SHE) formally by
then we may believe the stochastic integral of the last term would be martingale. However, [15] shows that the random field is singular with respect to Lebesgue measure. Therefore, the stochastic integral is formal. In our main results, we will show that it is indeed a martingale and its quadratic variation can be described in terms of .
1.1 Setting and known results
In this paper, we consider the model in the discrete setting as in [14].
Let be i.i.d. random variables with the law such that
| (1.1) |
Let be an irreducible, symmetric, and aperiodic random walk on whose increment has mean and covariance matrix being the identity matrix . Let and denote probability and expectation for . Also, we assume that has finite support, i.e. there exists a finite set such that .
We define the point-to-point partition function of -directed polymers in random environment
| (1.2) | |||
| (1.3) |
for () and , , where we use the convention for .
We note that
where we denote by the transition probability kernel of the underlying random walk starting at i.e.
for , .
For , we denote by the greatest integer . For , we define by the closest point of (if more than two points exist, we choose the smallest one in the lexicographic order). Also, we write and for and if it is clear from the context that and should be an integer and a lattice point.
For fixed , we focused on rescaled random measure valued flows which are defined by
for , where we take the summation over all . Then, we have for , , and
| (1.4) |
We define and by replacing by .
We equip with the space of locally finite measures on and finite measures on with the topology of vague convergence and the topology of weak convergence, respectively:
for and the locally finite measures on and
for and the finite measures on .
Remark 1.2.
To see the nontrivial limit of random measures obtained in [14, 15], we rescale the strength of disorders as properly.
Let be an independent copy of and be the expectation of the number of collisions of and up to time :
Then, the local limit theorem below gives that the asymptotic behavior
| (1.5) |
Theorem 1.3.
We choose the disorder strength such that
| (1.8) |
for some .
Theorem 1.4.
Remark 1.5.
We will see that in Remark 3.1, so also converges to .
Remark 1.6.
Remark 1.7.
In this paper, we focus only on the case (so ). Hence, we often omit the first coordinate of pair of times for a flow .
1.2 Measure valued process
We will show that for fixed ,
has a version which has continuous sample paths almost surely, where we define
and give a semimartingale representation.
First of all, we recall some facts about measure-valued process from [41] which is a textbook of Dawson-Watanabe superdiffusions (or super-Brownian motions).
Let be a Polish space. Then, we define
| the set of -valued paths with the topology of uniform convergence on compacts, | |||
| the set of càdlàg -valued paths with the Skorokhod -topology. |
Let be the set of finite measures on with the topology of weak convergence. Then, is also a Polish space [18, Theorem 3.1.7] and hence is a Polish space.
For a càdlàg -valued process , we denote by
| (1.9) |
the right-continuous filtration generated by a process .
Our first main result shows the existence of a continuous version of .
Theorem 1.9.
For each , there exists a continuous -valued process such that its finite dimensional distributions are identical to those of the Critical SHF.
Remark 1.10.
Our second theorem gives a martingale problem of the measure-valued process , which is similar to the form discussed in super-Brownian motion.
Theorem 1.11.
Let , , and . Let be the continuous process. We define
| (1.10) |
for . Then, is a continuous -martingale such that
| (1.11) |
where
| (1.12) |
and (1.11) is locally uniform convergence in probability.
Remark 1.12.
We remark that the martingale problem in Theorem 1.11 is ill-posed. Indeed, dropping the superscripts in (1.10)-(1.12) does not change the martingale problem. However, we find that
for by looking at their variances (see (2.13)). In particular, we know that the deterministic process
satisfies (1.11) with the constant quadratic variation. Thus, the martingale problem (1.10)-(1.12) has a family of solutions .
Remark 1.13.
Remark 1.14.
For a usual super-Brownian motions , their quadratic variation is given by
for some which is explicitly determined by the measure valued process .
Just on the one hand, the quadratic variation for super-Brownian motion with a single point catalyst is represented by the density field which is given by the limit of for [16].
By the definition of the cross variation of the martingales, we have the following.
Corollary 1.15.
Remark 1.16.
Theorem 1.11 gives the semimartingale representation of . It is natural to consider Itô’s formula to for a function . Then, we have
However, it is not obvious whether
| (1.15) |
in probability holds. The absolute continuity of in time with respect to the Lebesgue measure is not clear. It remains open.
Remark 1.17.
To formulate (KPZ) via Cole-Hopf transformation, we may look at instead of since the latter is ill-posed. Since for Lebesgue almost everywhere a.s. [15, (10.9)],
for Lebesgue almost everywhere a.s.. Thus, we also need to introduce another renormalization for this approach: Find and such that for each
1.2.1 Martingale measure
For fixed and , let be the set of continuous -martingales with and for each .
Then, we found that maps a function in to a process linearly.
In the following theorem, we will see the extension of the map .
Theorem 1.18.
Let , and . Then, there exists a unique linear extension of to such that
| (1.16) |
for each , where is the set of bounded Borel measurable functions and (1.16) is locally uniform convergence in probability.
For (), we denote by .
Remark 1.19.
Theorem 1.18 and Corollary 1.15 imply that would be an orthogonal martingale measure in the sense of Walsh [45, Chapter 2]. Indeed, for each , is a continuous martingale and if with , then
Moreover, we can define the stochastic integral with respect to this martingale measure in the general theory in [45]. However, we don’t discuss it in this paper.
1.3 Organization of the paper
In section 2, we review some results related to the analysis of moments of from [12, 15]. In section 3, we give an outline of the proof of Theorem 1.9 and Theorem 1.11. Section 4.1-6 are devoted to the detailed proofs. In section 7, we will discuss the regularity of the critical stochastic heat flow.
2 Variance and its limit
In this section, we will look at the variances of .
First, it is easy to see that
and
| (2.1) |
Also, the standard -moment method for DPRE yields
where and are independent random walks starting at and whose increments has the same law with , respectively. Since
| (2.2) |
we have
where we set and for and .
To see this quantity in detail, we use the following weighted local renewal functions introduced in [11] but we change the definition a little bit: For each ,
| (2.3) | ||||
| (2.4) | ||||
| and | ||||
| (2.5) | ||||
Then,
| (2.6) |
Thus, we can expect that plays a key role in controlling the modulus of continuity of .
We review some known results of and obtained in [11, 12]. We define
for , and we set
for and . In particular,
| (2.7) |
Theorem 2.1.
In this paper, we often omit the subscript and denote by
for and .
Proposition 2.2.
Theorem 2.3.
The higher moments of are given explicitly in [25, Theorem 1.1] and [15, Theorem 9.6]. To write it, we prepare some notations: Let be an integer. For (), we wefine
which is identified with . We denote by the integral of the integrable function on with respect to Lebesgue measure.
We define
for , and . Also, we define
for , and with .
Theorem 2.4.
Fix . Let be an integer. Then, for , , and
| (2.14) |
where we write and , and
3 Proofs of Theorem 1.9 and Theorem 1.11
3.1 as measure valued process
We fix .
Hereafter, we may assume that i.i.d. random variables with
| (3.1) |
Indeed, the convergence to is independent of the choice of satisfying (1.1). In this case, it is easy to see that are i.i.d. random variables with , where we set . In particular, we have
| (3.2) |
for .
(3.2) will help us estimating the chaos expansions of moments a bit (see Section 4), but it is not crucial.
We fix as in (1.8). For simplicity of the notation, we write
Then, can be regarded as a measure-valued process with the initial value
| and | ||||
We set
| for and | ||||
Now, we look at as a discrete semimartingale as in the construction of super-Brownian motion from critical branching Brownian motion [41, II. 4].
Let be a filtration generated by and we set for (). Then, we have
where we set for and .
We define a discrete Laplacian by
and we write
Then, we have
and hence
| (3.3) |
We remark that
is an -martingale with and the quadratic variation
In particular,
| (3.4) |
Remark 3.1.
By the same argument as above, we can see that
and hence
for each .
3.2 Continuity of
Definition 3.2.
Let be a Polish space. We say that the collection of processes with paths in is -relatively compact in if and only if it is relatively compact in and all weak limit points are a.s. continuous.
Definition 3.3.
Let be a Polish space. We say that is separating if and only if for any , for all implies .
Theorem 3.4.
[41, Theorem II.4.1] Let be a Polish space and be a separating class in containing . A sequence of càdlàg -valued processes is -relatively compact in if and only if the following conditions hold:
-
(1)
For all , , there exists a compact set in such that
-
(2)
For all , is -relatively compact in .
If in addition, is closed under addition, then the above equivalence holds when ordinary relative compactness in replaces -relative compactness in both the hypothesis and conclusion.
Coming back to , we take as the separating set in the proof of Theorem 1.9.
Proof of condition (1) in Theorem 3.4 for .
Fix and . Then, by the invariance pinciple, we can take a compact set such that
We regard as a measure on the path space of random walk by
Then, it is clear that
and the right-hand side is an -martingale. Therefore, we have
| (3.5) |
where we have used Doob’s maximal inequality in the second inequality. ∎
Next, we will verify (2) in Theorem 3.4 for . To see the -relative compactness in of , it is enough to see the following two conditions
-
(C-1)
the -relative compactness of in , and
-
(C-2)
the -relatively compactness of in .
One can show (C-1) by the following lemma.
Lemma 3.5.
For any and ,
Proof of (C-1).
Proof of Lemma 3.5.
Thus, the proof of Theorem 1.9 has been completed once one can verify (C-2). To prove (C-2), we apply the general theory for -tightness of martingales given in [41, Lemma II.4.5.] or the conclusion of [30, Theorem VI.4.13, Theorem VI.3.26]
Lemma 3.6.
Let be martingales with . Let
and extend and to as right-continuous step functions.
Then, the followings hold:
-
Suppose the following two conditions:
-
is -relatively compact in .
-
(3.7)
Then. is -relatively compact in .
-
-
If, in addition,
(3.8) then implies that is a continuous -martingale with respect to the filtration and that
3.2.1 Proof of for
To prove in Lemma 3.6 for , we adapt the standard method.
Lemma 3.7.
For each , , , and , there exists such that
| (3.9) |
Then, applying the Garsia-Rodemich-Rumsey inequality [22, Theorem A.1], follows.
Remark 3.8.
Here is an easy estimate of the difference between the values of . For ,
| (3.10) |
where we have used for in the first term, and for and
in the second term.
3.2.2 Proof of for
To prove (3.7), we use the following Burkholder type inequality [41, (PSF) in p.152] and [4, Theorem 21.1].
Lemma 3.9.
Let be a continuous increasing function with such that there exists such that for any .
Let be an -martingale. We set ,
Then, we have
Proof of (3.7) for .
Conditioned on , is a sum of mean independent random variables , where
Let be the subset of such that , for and . We define the filtration . Then,
is -martingale, and since the summation is finite. Thus, we can apply Lemma 3.9 with () to . Also, we can see that
by (3.1). Thus, we obtain
where we have used in the first inequality.
We can see from [12, Lemma 6.1 (6.4)] that for any and , there exists such that
| (3.12) |
∎
Remark 3.10.
For (3.12), we see that
Our setting of , in particular (3.2), allows us to use [12, Lemma 6.1 (6.4)] for an upper bound of the first term. More precisely, the expectation was divided into two terms, non-triple intersections and triple intersections, and the latter one vanishes under (3.2). The second term can be easily estimated by (2.6) and Lemma 2.1.
3.2.3 Proof of for
Proof of in Lemma 3.6.
We use Lemma 3.9 again. Taking ,
The second term is already estimated in the proof of . Also, the expectation in the first term is dominated by
from [4, Lemma 16.1]. Also, combining Theorem 15.1 in [4] and Doob’s -inequality, this is dominated by
where we remark that if , then . We know that the right-hand side is bounded from [12, Theorem 1.4]. ∎
3.3 Martingale
Suppose that satisfy all conditions in Lemma 3.6 so Theorem 1.9 follows. Theorem 1.4 implies that in .
By the Skorokhod representation theorem, we may assume that and are defined on a common probability space and in a.s.
Then, all terms in the righthand side of (3.3) (and hence ) converge almost surely. Indeed, , and Taylor’s theorem implies that for each and , there exists such that
where is the Hesse matrix of .
Thus, we found that in and hence, Lemma 3.6 implies that
| (3.13) |
and that is a continuous -martingale (not -martingale).
Therefore, the proof of Theorem 1.11 is completed when we proved the following two lemmas.
Lemma 3.11.
For any , , and , is a continuous -martingale.
Lemma 3.12.
For any , , and ,
for any .
Also, we give a corollary on the quadratic variation.
Corollary 3.13.
For any , , , and
Proof.
Since we proved that is uniform integrable, we have
Then, we can use the same argument in the proof of Theorem 1.2 in [12]. ∎
3.3.1 Proof of Lemma 3.11
Since is a separating set for ,
and for .
Let , and be a bounded continuous function on . Then, we have
The uniform integrability of and implies
Thus, we completed the proof of Lemma 3.11.
3.3.2 Proof of Lemma 3.12
Since we know that for each and
Fatou’s lemma implies that
We will prove the following lemma.
Lemma 3.14.
We have
| (3.14) |
for , , and .
Proof of Lemma 3.12.
The proof of Lemma is postponed to Section 6.
We will verify the convergence of expectations of quadratic variation as an exercise and give the proof of existence of extension in Theorem 1.18: Theorem 2.3 and Lemma 3.6 imply that
| (3.15) |
Also,
where we have used that . Since , we have
where we have used the following lemma and the dominated convergence theorem in the last equation.
Lemma 3.15.
Let and . Then, for each , there exists such that
| (3.16) | ||||
| for each and | ||||
| (3.17) | ||||
for a.e. for .
Proof.
It is easy to see that
for some .
Also,
Therefore, we obtain from l’Hôpital’s rule that
if the limit in the righthand side exists for each .
It is easy to see that it should be equal to
a.e. by Lebesgue’s differential theorem. Therefore, (3.17) holds for . ∎
Proof of existence of extension in Theorem 1.18.
Let be a bounded Borel function. Then, Lusin’s theorem and Tietze extension theorem implies that there exists a sequence in such that
(See [43, Remark 1.3.30].) Also, we know any bounded continuous function can be approximated by uniformly on any compact sets.
Hence, we can choose such that
| (3.18) |
For each , we set satifying (3.18). Then, we can see from the Burkholder-Davis-Gundy inequality and (3.15) that for , and
| (3.19) |
for some constant . Thus, the dominated convergence theorem implies that is an -Cauchy sequence and -tight and hence the limit exists and we denote it by .
∎
The proof for the representation of the quadratic variations of will be given in Section 6. However, the above proof implies the following convergence of the quadratic variation.
Corollary 3.16.
Let , . Then, for each and ,
where is a sequence in such that converges to for any and .
Proof.
Remark 3.17.
Combining Theorem 2.3 and Doob’s inequality, we can find that the sequence of process weakly converges to a process .
The Skorkhod representation theorem allows us to has the same form (1.10).
Remark 3.18.
We remark that every term in expectation in (3.9) and (3.14) can be described via partition functions from (3.4). More precisely, we need look at
| (3.21) |
for Lemma 3.7, and the linear combination of
| (3.22) | |||
| (3.23) | |||
| (3.24) |
for Lemma 3.14.
Thus, we will entirely focused on computing of moments of partition functions in the following sections.
4 Moments of partition functions
From now, we will omit the parameter in the notations if it is clear from the context.
4.1 Chaos expansion and moments
Let be a countable set and be independent Bernoulli distributed random variables with .
For a finite subset , we define
and the polynomial chaos is defined as a linear combination of , i.e.
for some such that except for some . We set for convention.
Then, it is easy to see that if any belongs to exactly even number of subsets , and is equal to otherwise.
Now, we will give the polynomial chaos expansion of partition functions: Take . For , , ,
where for with (), we define
and
Here, we define for our convention and we set
For our convenience, we introduce a set of finite subsets of indices
for , where we set for . We define .
The above argument gives that
| (4.1) |
Now, we focus on the finite subsets that contribute to the summation in the right-hand side.
We say have an odd intersection if one of the following holds:
-
(1)
there exists an such that belongs to three of but does not belong to the other one.
-
(2)
there exists an such that belongs to one of but does not belong to the others.
Also, we say have even intersections if don’t have an odd intersection.
We have from (3.2)
Thus, that have even intersections contribute to the summation of in (4.1).
The above argument yields that
| (4.2) |
We can write (3.22)-(3.24) in similar ways, but we omit giving them here.
Hereafter, we may assume that have even intersection.
4.2 Pairings of intersections
We write elements of in time ordered as
| (4.3) |
where for the case , we may choose such that since if for some , then belong to two of and belongs to the other two). Also, we define by
the sequence of intersection times.
Definition 4.1.
Suppose with (4.3). If have even intersections and , for each , one of the following three cases occurs:
-
( for )
-
belongs to two of but not to the other two. Moreover, for any .
-
.
-
-
( for ) and belongs to two of and belongs to the other two.
Thus, when has even intersections, each has an associated pair(s) of indices, denoted by , (), (), (), where .
We set . Then, we define the associated map which maps to a finite -sequence if have even intersection and :
| (4.4) |
Definition 4.2.
We say are a couple if , i.e. each of , , and is a couple.
We denote by the set of finite sequences of . Then, (4.2) is rewritten by
| (4.5) |
Next, we will see that the contributions to (4.2) from and can be identified with the contribution from .
For fixed , we consider the contributions from and to the summation in spatial variables at .
The contribution from at to (4.2) has the form of
| (4.6) | |||
Also, we can see from (3.2) that the contribution from at to (4.2) has the form that replaces by in the summation of (4.6).
Thus, we can identify the contribution from at to (4.2) with the one from . This is one-to-one correspondence.
Hence, we may consider that the summations in (4.5) of are taken over the finite sequence in .
By a similar way, we can see that the contributions from are identified with the one from . This is one-to-one correspondence.
We write by the map from to deduced from the above correspondence. We denote the length of by .
Then, (4.5) can be rewritten by
| (4.7) |
For , we can see that
| (4.8) |
where is the set of time-sequence given as follows:
-
(T-1)
are associated with for .
-
(T-2)
.
-
(T-3)
If (or ), then (). Otherwise, .
-
(T-4)
If and is not a couple, then . Otherwise is allowed.
Remark 4.3.
does not appear in the sum on the right-hand side explicitly. However, contains all their information.
Remark 4.4.
The inequality comes from the fact that is mapped to so that the time-space summation associated with does not contain the quadruple intersection. However, the difference is negligible. Indeed, the differences is dominated from above by the summation of quadruple intersection terms of expansion of . However, we can find from the proof of [14, Theorem 6.1] that the contribution from the quadruple intersections is negligible (see the argument after Proposition 6.6 in [14]).
4.3 Partitions of sequence of pairings by stretches
Next, we focus on consecutive sequences in , called stretches in [12], that is, can be divided into some blocks , where we define
-
(1)
.
-
(2)
For each , if . Otherwise, we define .
Thus, each block is associated with an element of . We denote the number of stretches in by .
We denote by the set of finite sequences of with ().
We define the map from to by .
Then, we can find that
Also, we can see that
where is the set of space-time-sequence given as follows:
-
(-1)
are associated with the stretch for , which represents the start point and the end point of the stretch.
-
(-2)
.
-
(-3)
If (), then (). Otherwise, .
-
(-4)
If and is not a couple, then . Otherwise is allowed.
-
(-5)
.
Also, for , and , is the set of space-time-sequence given as follows:
-
(-1)
are associated with for .
-
(-2)
.
-
(-3)
.
Now, we focus on the summation over , and . Fix and . Then, the other variables appear in the summand with the form
where is a function independent of the summation. Hence, it has the given by
Repeating this procedure, we rewrite (4.8) by the following form:
To give the explicit form of , we will see the sequence of , .
Definition 4.5.
For each and , we set
where we set . Also, we denote by the number of times appears in .
Also, we set
for , , and , and .
For each , the transition between and is from to . Its contributions to are given by the form for some . Thus, we can see that
Moreover, we remark that the summation
can be embedded into the summation of and . Hence, we have
| (4.9) |
where we set
and is the set of sequence of time-pairs satisfy the followings:
-
(-1)
are associated with the stretch for , which represents the start time and the end time of the stretch.
-
(-2)
.
-
(-3)
.
-
(-4)
If and are not a couple, then . Otherwise, is allowed.
In particular, the summand is given as the products of the weights associated with the graphs.
Thus, it is enough to estimate (4.9).
We now introduce a new oriented graph with vertices , where the oriented edges are () and for ().
We write
| and | ||||
Then,
| (4.10) |
We can see the following structure of the graph .
Definition 4.6.
Each has two incoming edges , and two outgoing edge and , where and .
Let .
For simplicity, we set , , and . (The other cases are obtained by permutation.)
Proposition 4.7.
For each , the following holds.
-
(1)
. Also, the following holds:
-
(i)
If and are a couple, then and .
-
(ii)
If and are not a couple, then , , , , and . In particular, if and only if and are a couple.
-
(i)
-
(2)
Let .
-
(i)
If and are a couple, then and the equality holds if and only if .
-
(ii)
If and are not a couple, then and . In particular, if and only if each label in is contained in either or .
-
(i)
-
(3)
For , , .
-
(4)
Let . If and are a couple, then . If and are not a couple, and . In particular, if and only if each label in are contained in or
Proof.
(1) is trivial by definition.
(1)(i) If and are a couple, then so that and hence .
(1)(ii) If and are not a couple, then (). So and there exists oriented edges and . Also, it is trivial that . Finally, if and are a couple (e.g. and ), then for since it does not contain and . Thus, . On the other hand, if and are not a couple, then holds by definition.
(2)(i) If and are a couple (e.g. and ), then for and since . Therefore, . Also, if and only if there exist such that and , i.e. .
(2)(ii) If and are not a couple (e.g. and ), then and . If for , then exists. On the other hand, if for , then does not exist so .
(3) By definition, the labels contained in are . Then, for , and are not a pair(e.g. and ), and and are not a pair and hence or .
∎
Now, we will give an upper bound of (4.9) by taking summation in spatial variables .
First, we will take summation in the order of as follows: We remark that () appear in just one time and in and for just two and the same holds for ().
The summand of (4.10) has the form , and hence the summation of (4.10) in ) is dominated by . In particular, appears as in .
We know that for ,
| (4.11) |
if , where is a constant which is uniformly chosen in and ,
| (4.12) |
if , where is a constant depending only on , and
| (4.13) |
if , where is a constant depending only on . We denote by
Hence, and appear in the summand with the form (if ) or (if ).
Let . Suppose that by taking summation in and , the summand has the form
| (4.14) | |||
| (4.18) |
Since appears only in , the summation in of (4.14) is dominated by
| (4.19) |
and appears as in (4.19). The summation of (4.19) in is dominated by
By induction, we can obtain an upper bound of (4.10). To give it, we divide into two disjoint sets
For , one (4.13) and two (4.12) appear. On the other hand, for , two (4.13) and no (4.12) appear.
Thus, we can find that (4.10) is dominated by
| (Type-) | |||
| (Type-) |
Here is a constant (chosen later) which will play the same role as the one introduced in [12].
![[Uncaptioned image]](https://cdn.awesomepapers.org/papers/2b5e1a1e-84ec-442c-8b0a-666dc233f245/x2.png)
![[Uncaptioned image]](https://cdn.awesomepapers.org/papers/2b5e1a1e-84ec-442c-8b0a-666dc233f245/x3.png)
For (Type-), we have four cases
-
(Type-1)
-
(Type-2)
-
(Type-3)
-
(Type-4)
and for (Type-), we have three cases
-
(Type-6)
-
(Type-7)
-
(Type-8)
.
Also, we will divide the summation by the first site after .
Definition 4.8.
For each sequence in , there exists an such that one of the following holds:
-
-
-
-
.
5 Bounds of moments
To give upper bounds of moments, we use some estimates in [12], where they gave upper bounds of third moments in terms of multivariate integrals. Essentially, the method is the same as the one in [12] but our integrands are complicated.
5.1 -(Type-1)-case
5.1.1 Change of time variables
At first, we will change of time variables as follows.
For a while, we fix .
We change the time sequence as follows:
Then, we replace the variables of the summation from to and enlarge the range of them as follows
Since and , we can see that
| (5.1) | ||||
| and | ||||
| (5.2) | ||||
We use the following result.
Lemma 5.1.
[12, Lemma 5.3 and (5.37)] For each and , there exists a constant such that for any
Corollary 5.2.
For each and , there exists a constant such that for any
Now, we focus on
which is the summation of the product of terms with respect to variables. By the AM-GM inequality for , , it is dominated by
| (5.5) |
Now, we integrate it in order from to .
Since appear as in integrand of (5.5),
To estimate the integral in the right-and side, we use the results in [12].
We define
Then, it is easy to see that
Lemma 5.3.
In particular,
for and .
Therefore,
| (5.6) |
for .
Also, we have
| (5.7) |
5.2 -(Type-2)-case
For a while, we fix .
We change the time sequence as follows:
Then, we replace the variables of the summation from to and enlarge the range of them as follows:
Using (5.1), we can see that
| (5.8) |
where we remark that since must be larger than . Also, we remark that the summand does not contain .
Then, a similar argument to the analysis of (5.5) yields that
The rest of analysis is almost the same as the one of the proof after (5.5), so we omit it.
Anyway, we can obtain that
for large enough, where is a constant depending on .
6 Proof of Lemma 3.14
The limit of the third moment of was obtained in [12] and the higher moments of the moments in the continuous setting (stochastic heat equation) was obtained in [25].
We recall that for each and , we set
where we set . Also, we denote by the number of times appears in .
Also, we define
for , , , and , where we set , , , and for and .
Lemma 6.1.
Let and . For each ,
| (6.1) |
Lemma 6.2.
Let and . For each ,
| (6.2) |
Lemma 6.3.
Let and . For each ,
| (6.3) |
We give an outline of the proof of Lemma 6.2 and omit the proofs of Lemma 6.1 and 6.3 since the argument are almost the same.
Remark 6.4.
6.1 Proof of Lemma 6.2
We first give the chaos expansion of the expectation of
in a similar way to the argument in Section 4. We use the label derived from the “random walks” in and derived from the “random walks” in .
We remark that after the last intersection between and , each of them may meet or but after the last intersection between and , neither nor will meet other particles. Thus, contributing the chaos expansion should satisfy one of
-
(1)
-
(2)
and .
As we mentioned in Remark 4.4, quadruple intersections in the chaos expansion of moments are negligible. Therefore, we have the following representation of the moment. We omit its proof since it is almost the same as the discussion in (4.9).
Lemma 6.5.
Let and . For each , we have
| (6.5) |
where is the set of time sequences which satisfy the followings:
-
(-1)
are associated with the stretch for , which represents the start time and the end time of the stretch.
-
(-2)
.
-
(-3)
If and are not a couple, then . Otherwise, is allowed.
Also, we set
where we write , , , and .
Thus, it is enough to see the limit of the right-hand side of (6.5) for the proof of Lemma 6.2. Here, we give an idea of the proof of this convergence since it is almost the same as the proof of [12, (5.3)].
Indeed, we may regard it as “Riemannian summation” for some function due to the following approximation:
In particular, we can find that the approximations of and yield the factor and , respectively, so we obtain the factor .
To prove the convergence, we first look at the second term of (6.5) with the near diagonal sets are cut off, i.e.
for some fixed . It approximates (6.5) uniformly in since we know that the boundedness of the fourth moment of and the second moment of .
Also, we can find from Remark 6.4 that the second term of (6.2) is approximated by the one restricted by
Similarly to the arguments in [12, (5.3)], we need to consider the sumations or integrals with the restricted spatial variables and to the set
| for (6.2) | |||
| for (6.5) |
for large .
Proof of Lemma 3.14.
Thus, it is enough to see
as .
For (6.1), we first remark that
| (6.8) |
for and some . Moreover, we find that
by Lemma 3.15. It does hold for .
In particular, they converge to and if and only if and ( or ).
The dominated convergence theorem yields
Applying the same argument to (6.2), we obtain that
∎
Remark 6.6.
In Theorem 1.11, quadratic variation is approximated by using . However, we can approximate it by using with satisfying .
Indeed, we can modify the proof by using the following lemma instead of Lemma 3.15
Lemma 6.7.
Let , , and . Then, for each , there exists such that
| (6.9) |
and
| (6.10) |
for each .
7 Peaks of
It is known that is singular with respect to Lebesgue measure [15, Theorem 10.5]. It follows from the fact that converges to for Lebsegue a.e. ([15, (10.9)]).
Thus, it follows that there exists such that
Furthermore, [15, Theorem 10.6] says that for any and belongs to , where is the negative Besov-Hölder space of order in the sense of [21, Definition 2.1].
In this section, we will give a “typical order of peak” of .
Fix with . We define a random set
where is a Borel set with finite Lebsegue measure and .
Theorem 7.1.
Fix and with . Let be an open set and .
-
(1)
We have
-
(2)
There exists a non-random decreasing sequence with such that
We can find that the peaks of are of order . Thus, we may expect that belongs to the “logarithmic HaiHölder space” , where we say belongs to for and or and if belongs to the dual of with and
Remark 7.2.
The reader may refer to [28, Definition 3.7] for the definition of and . In [28], he referred the relationship between and the Besov space . Then, is a slight modification of . To our knowledge, there are no results concerning with the relationship between and the generalized Besov space (discussed in e.g. [37, 20, 1]).
We have the following result.
Theorem 7.3.
Fix , and with . Then, is not a continuous -valued process with the uniform-on-compact topology for any .
Proof of Theorem 7.1.
Then, it is easy to see that
for any with positive probability and hence it is a contradiction.
Let be an open set with . Let be a positive function such that
We focus on the martingale . We can see
Indeed, we have
from the Paley-Zygmund inequality. Moreover, it follows from Lemma 3.14, (3.13), and Fatous’s lemma that the dominator in the right-hand side is bounded. Also, we can find from Corollary 3.13 that the numerator is strictly positive.
We set
We remark that if for some , then is not empty set.
Thus, it is enough to prove that
for .
Since we have for ,
Since the right-hand side is bounded for and , there exists a random such that
on , and hence, it follows that
∎
It is natural to expect that the values of of order contributes to and . That is, we may expect
for large enough. Let
Then, we would find that
for large .
Acknowledgemments This work was supported by JSPS KAKENHI Grant Number JP22K03351, JP23K22399. The author thanks Prof. Nikos Zygouras for useful comments.
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